A307164 -id:A307164 - cp's OEIS frontend

A307163 Minimum number of intercalates in a diagonal Latin square of order n.

0, 0, 0, 12, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Eduard I. Vatutin, Mar 27 2019

An intercalate is a 2 X 2 subsquare of a Latin square.
Every diagonal Latin square is a Latin square, so 0 <= a(n) <= A307164(n) <= A092237(n). - Eduard I. Vatutin, Sep 21 2020
Every intercalate is a partial loop and every partial loop is a loop, so 0 <= a(n) <= A307170(n) <= A307166(n). - Eduard I. Vatutin, Oct 19 2020
a(n)=0 for all orders n for which cyclic diagonal Latin squares exist (see A007310) due to all cyclic diagonal Latin squares don't have intercalates. - Eduard I. Vatutin, Aug 07 2023
a(n)=0 for all orders n for which diagonalized cyclic diagonal Latin squares exist (see A372922) due to all diagonalized cyclic diagonal Latin squares don't have intercalates. - Eduard I. Vatutin, Sep 24 2024
a(16) <= 2, a(17) = 0, a(18) <= 9, a(19) = 0, a(20) <= 1, a(21) <= 11, a(22) <= 9, a(23) = 0, a(24) <= 16, a(25) = 0, a(26) <= 29. - Eduard I. Vatutin, added Sep 10 2023, updated Mar 01 2025

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
E. I. Vatutin, About the minimum number of intercalates in a diagonal Latin squares of order 9 (in Russian)
E. I. Vatutin, On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops (in Russian)
Eduard I. Vatutin, About the heuristic approximation of the spectrum of number of intercalates in diagonal Latin squares of order 14 (in Russian)
Eduard I. Vatutin, About the minimum number of intercalates in diagonal Latin squares of order 15 (in Russian)
E. Vatutin, A. Belyshev, N. Nikitina, and M. Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, Communications in Computer and Information Science, Vol. 1304, Springer, 2020, pp. 127-146, DOI: 10.1007/978-3-030-66895-2_9.
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021). Tula, 2021. pp. 7-17. (in Russian)
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A007310, A307164, A092237, A307170, A307166, A345760, A372922.

a(9) added by Eduard I. Vatutin, Sep 21 2020
a(10)-a(13) added by Eduard I. Vatutin, Apr 01 2021
a(14)-a(15) added by Eduard I. Vatutin, Sep 24 2024

A345760 a(n) is the number of distinct numbers of intercalates of order n diagonal Latin squares.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 21, 61, 64

Offset: 1

Eduard I. Vatutin, Jun 26 2021

a(n) <= A307164(n) - A307163(n) + 1.
a(n) <= A287764(n).
a(10) >= 98, a(11) >= 145, a(12) >= 259, a(13) >= 200, a(14) >= 362, a(15) >= 536, a(16) >= 792, a(17) >= 685, a(18) >= 535, a(19) >= 447, a(20) >= 1011, a(21) >= 747, a(22) >= 872, a(23) >= 885, a(24) >= 1610, a(25) >= 1677, a(26) >= 1266, a(27) >= 1337, a(28) >= 2795. - Eduard I. Vatutin, Oct 02 2021, updated Mar 02 2025

			For n=7 the number of intercalates that a diagonal Latin square of order 7 may have is 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 26, or 30. Since there are 21 distinct values, a(7)=21.

Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of orders 1-7 (in Russian).
Eduard I. Vatutin, About the spectra of numerical characteristics of diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, About the approximation of spectra of numerical characteristics of diagonal Latin squares of order 9 (in Russian).
Eduard I. Vatutin, About the results of experiment with spectra of diagonal Latin squares using Brute Force and distributed computing projects Gerasim@Home and RakeSearch (in Russian).
Eduard I. Vatutin, Graphical representation of the spectra.
Eduard I. Vatutin, Proving lists (1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, I. I. Kurochkin, A. M. Albertyan, A. V. Kripachev, A. I. Pykhtin, Methods for getting spectra of fast computable numerical characteristics of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 19-23. (in Russian)
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan and I. I. Kurochkin, Estimation of the Cardinalities of the Spectra of Fast-computable Numerical Characteristics for Diagonal Latin Squares of Orders N>9 (in Russian) // Science and education in the development of industrial, social and economic spheres of Russian regions. Murom, 2022. pp. 314-315.
E. I. Vatutin, V. S. Titov, A. I. Pykhtin, A. V. Kripachev, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, I. I. Kurochkin, Heuristic method for getting approximations of spectra of numerical characteristics for diagonal Latin squares, Intellectual information systems: trends, problems, prospects, Kursk, 2022. pp. 35-41. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A287764, A307163, A307164, A309344, A344105, A345370, A345761.

a(9) added by Eduard I. Vatutin, Oct 22 2022

A307166 Minimum number of loops in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 12, 10, 27, 21, 40

Offset: 1

Eduard I. Vatutin, Mar 27 2019

A loop in a Latin square is a sequence of cells v1=L[i1,j1] -> v2=L[i1,j2] -> v1=L[i2,j2] -> ... -> v2=L[im,j1] -> v1=L[i1,j1] of length 2*m that consists of a pair of values {v1, v2}.
For diagonal Latin squares of order 4 all loops are intercalates. - Eduard I. Vatutin, Oct 05 2020
From Eduard I. Vatutin, Oct 26 2020: (Start)
Every intercalate is a partial loop and every partial loop is a loop, so 0 <= A307163(n) <= A307170(n) <= a(n).
0 <= a(n) <= A307167(n).
(End)

			For example, the square
  2 4 3 5 0 1
  1 0 4 3 2 5
  0 2 5 4 1 3
  5 3 0 1 4 2
  4 5 1 2 3 0
  3 1 2 0 5 4
has a loop
  2 4 . . . .
  . . . . . .
  . 2 . 4 . .
  . . . . . .
  4 . . 2 . .
  . . . . . .
consisting of the sequence of cells L[1,1]=2 -> L[1,2]=4 -> L[3,2]=2 -> L[3,4]=4 -> L[5,4]=2 -> L[5,1]=4 -> L[1,1]=2 with length 6.
The total number of loops for this square is 21.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the minimum and maximum number of loops in a diagonal Latin squares of order 8 (in Russian).
E. I. Vatutin, On the inequalities of the minimum and maximum numerical characteristics of diagonal Latin squares for intercalates, loops and partial loops (in Russian).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307163, A307164, A307167, A307170.

a(8) added by Eduard I. Vatutin, Oct 05 2020

A307167 Maximum number of loops in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 12, 14, 27, 53, 112

Offset: 1

Eduard I. Vatutin, Mar 27 2019

A loop in a Latin square is a sequence of cells v1=L[i1,j1] -> v2=L[i1,j2] -> v1=L[i2,j2] -> ... -> v2=L[im,j1] -> v1=L[i1,j1] of length 2*m that consists of a pair of values {v1, v2}.

For diagonal Latin squares of order 4 all loops are intercalates. - Eduard I. Vatutin, Oct 05 2020

			For example, the square
  2 4 3 5 0 1
  1 0 4 3 2 5
  0 2 5 4 1 3
  5 3 0 1 4 2
  4 5 1 2 3 0
  3 1 2 0 5 4
has a loop
  2 4 . . . .
  . . . . . .
  . 2 . 4 . .
  . . . . . .
  4 . . 2 . .
  . . . . . .
consisting of the sequence of cells L[1,1]=2 -> L[1,2]=4 -> L[3,2]=2 -> L[3,4]=4 -> L[5,4]=2 -> L[5,1]=4 -> L[1,1]=2 with length 6.
The total number of loops for this square is 21.

Eduard I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
Eduard I. Vatutin, About the minimum and maximum number of loops in a diagonal Latin squares of order 8 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307166, A307163, A307164.

a(8) added by Eduard I. Vatutin, Oct 05 2020

A307839 Minimum number of Latin subrectangles in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 137, 336, 884, 1968, 4545

Offset: 1

Eduard I. Vatutin, May 01 2019

An Latin subrectangle is a m X k Latin rectangle of a Latin square of order n, 1 <= m <= n, 1 <= k <= n.

			For example, the square
  0 1 2 3 4 5 6
  4 2 6 5 0 1 3
  3 6 1 0 5 2 4
  6 3 5 4 1 0 2
  1 5 3 2 6 4 0
  5 0 4 6 2 3 1
  2 4 0 1 3 6 5
has a Latin subrectangle
  . . . . . . .
  . . 6 5 0 1 3
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . 0 1 3 6 5
The total number of Latin subrectangles for this square is 2119.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the minimum and maximum number of Latin subrectangles in a diagonal Latin squares of order 8 (in Russian).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307163, A307164, A307840, A307841, A307842.

a(8) added by Eduard I. Vatutin, Oct 06 2020

A307840 Maximum number of Latin subrectangles in a diagonal Latin square of order n.

Original entry on oeis.org

1, 0, 0, 137, 348, 884, 2119, 5433

Offset: 1

Eduard I. Vatutin, May 01 2019

An Latin subrectangle is a m X k Latin rectangle of a Latin square of order n, 1 <= m <= n, 1 <= k <= n.

			For example, the square
  0 1 2 3 4 5 6
  4 2 6 5 0 1 3
  3 6 1 0 5 2 4
  6 3 5 4 1 0 2
  1 5 3 2 6 4 0
  5 0 4 6 2 3 1
  2 4 0 1 3 6 5
has a Latin subrectangle
  . . . . . . .
  . . 6 5 0 1 3
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . 0 1 3 6 5
The total number of Latin subrectangles for this square is 2119.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).
E. I. Vatutin, About the minimum and maximum number of Latin subrectangles in a diagonal Latin squares of order 8 (in Russian).
Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A274806, A307163, A307164, A307839, A307841, A307842.

a(8) added by Eduard I. Vatutin, Oct 06 2020

A360223 Maximum number of intercalates in an orthogonal diagonal Latin square of order n.

Original entry on oeis.org

0, 0, 0, 12, 0, 0, 18, 112, 72

Offset: 1

Eduard I. Vatutin, Jan 30 2023

An intercalate is a 2 X 2 subsquare of a Latin square.
An orthogonal diagonal Latin square is a diagonal Latin square that has at least one orthogonal diagonal mate.
a(10) >= 76, a(11) >= 94, a(12) >= 324, a(13) >= 26. - Eduard I. Vatutin, updated Feb 25 2024

Eduard I. Vatutin, About the spectra of numerical characteristics of orthogonal diagonal Latin squares of orders 1-11 (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
E. I. Vatutin, N. N. Nikitina, M. O. Manzuk, A. M. Albertyan, and I. I. Kurochkin, On the construction of spectra of fast-computable numerical characteristics for diagonal Latin squares of small order, Intellectual and Information Systems (Intellect - 2021), Tula, 2021, pp. 7-17. (in Russian)
Index entries for sequences related to Latin squares and rectangles.

Cf. A305570, A305571, A307164, A354050, A360221.

A382952 Maximum number of intercalates in an extended self-orthogonal diagonal Latin square of order n.

Original entry on oeis.org

0, 0, 0, 12, 0, 0, 18, 112, 72, 53

Offset: 1

Eduard I. Vatutin, Apr 09 2025

A self-orthogonal diagonal Latin square (SODLS) is a diagonal Latin square orthogonal to its transpose. An extended self-orthogonal diagonal Latin square (ESODLS) is a diagonal Latin square that has an orthogonal diagonal Latin square from the same main class. SODLS is a special case of ESODLS.
Table showing minimums and maximums:
  order                      |  4  5  6  7   8   9  10
  min number of intercalates | 12  0  -  0  16   0   5
  max number of intercalates | 12  0  - 18 112  72  53 (this sequence)

Eduard I. Vatutin, About the properties of ESODLS of orders 1-10 (in Russian).
Eduard I. Vatutin, Proving list, minimum number of intercalates (best known examples).
Eduard I. Vatutin, Proving list, maximum number of intercalates (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A092237, A307164, A309210, A309598, A309599, A360223, A382957.

A382270 Maximum number of intercalates in a Brown's diagonal Latin square of order 2n.

Original entry on oeis.org

0, 12, 9, 112, 57

Offset: 1

Eduard I. Vatutin, Mar 20 2025

A Brown's diagonal Latin square is a horizontally symmetric row-inverse or vertically symmetric column-inverse diagonal Latin square (see A339641).
Brown's diagonal Latin squares are special case of plain symmetry diagonal Latin squares that do not exist for odd orders.
a(6)>=252, a(7)>=385, a(8)>=960, a(9)>=329, a(10)>=356, a(11)>=497, a(12)>=1008, a(13)>=497, a(14)>=524.

Eduard I. Vatutin, About the spectra of numerical characteristics of Brown's diagonal Latin squares (in Russian).
Eduard I. Vatutin, Proving list (best known examples).
Index entries for sequences related to Latin squares and rectangles.

Cf. A092237, A307163, A307164, A339641, A345760, A368182, A379665.

A336764 Maximum number of order 3 subsquares in a Latin square of order n.

Original entry on oeis.org

0, 0, 1, 0, 0, 4, 7

Offset: 1

Omar Aceval Garcia, Cameron Byer, Eugene Fiorini, Nicholas Hanson, Brian G. Kronenthal, Lindsey Wise, Aug 03 2020

A subsquare of a Latin square is a submatrix (not necessarily consisting of adjacent entries) which is itself a Latin square. (I. M. Wanless, Latin Squares with One Subsquare, Wiley and Sons)

R. Bean, Critical sets in Latin squares and Associated Structures, Ph.D. Thesis, The University of Queensland, 2001.
K. Heinrich and W. Wallis, The Maximum Number of Intercalates in a Latin Square, Combinatorial Math. VIII, Proc. 8th Australian Conf. Combinatorics, 1980, 221-233.
I. M. Wanless, Latin Squares with One Subsquare, Journal of Combinatorial Designs, 9 (2001), 128-146.

Cf. A092237, A091323, A090741, A307163, A307164.

a(3^n) = 9*a(3^(n-1)) + 27^(n-1) (conjectured).

cp's OEIS Frontend

A307163 Minimum number of intercalates in a diagonal Latin square of order n.

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A345760 a(n) is the number of distinct numbers of intercalates of order n diagonal Latin squares.

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A307166 Minimum number of loops in a diagonal Latin square of order n.

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A307167 Maximum number of loops in a diagonal Latin square of order n.

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A307839 Minimum number of Latin subrectangles in a diagonal Latin square of order n.

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A307840 Maximum number of Latin subrectangles in a diagonal Latin square of order n.

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A360223 Maximum number of intercalates in an orthogonal diagonal Latin square of order n.

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A382952 Maximum number of intercalates in an extended self-orthogonal diagonal Latin square of order n.

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A382270 Maximum number of intercalates in a Brown's diagonal Latin square of order 2n.

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A336764 Maximum number of order 3 subsquares in a Latin square of order n.

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